Analytical Upward Continuation of the Field

Here C is some constant. Thus, the expression for the potential is 

We have derived a formula for calculation of the potential of the field everywhere in the upper half space if both the potential U and the vertical component of the attraction field are known at the earth s surface, iS o- Having taken the derivative from both sides of Equation (1.110) we obtain for the vertical component of the attraction field at the point p  

Indeed, it satisfies Laplace s equation everywhere except at the point p, since it describes up to a constant the potential of a point mass located at the point p. Also, it has a singularity at this point and provides a zero value of the surface integral over the hemisphere when its radius r tends to infinity. Correspondingly, we can write 

However, this expression is impractical since the potential U is not measured on the earth s surface, and therefore we have to choose a Green s function such that the derivative dGfdz on the earth s surface would be zero. In this case Equation (1.110) is greatly simplified  

in the upper space, (z 0), Green s function is a solution of the Laplace s equation everywhere except at the observation point p  

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